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What Is Half-Life? Formula and Meaning

Physics · nuclei · NEET

Half-life (T or T-half) is the time in which half of the radioactive nuclei in a sample decay. It is fixed for a given element and does not depend on how many atoms you start with. Formula: T = 0.693 / lambda (where lambda is the decay constant). Memory hook: "Half gone every half-life" - after each half-life the amount left keeps halving: full, then half, then quarter, then one-eighth.
Half-Life: Amount Left Halves Every TN0N0/2N0/4N0/801 T2 T3 TTime (in units of half-life T)100%50%25%12.5%
Each half-life T removes half of what remains: 100% to 50% to 25% to 12.5%. The amount left after n half-lives is (1/2)^n times the starting amount, so it keeps halving and never reaches zero.

Your doubts, answered

Does half-life change if I start with more atoms?

No. Half-life is a fixed property of the nucleus. Whether you start with 1000 atoms or 1 crore atoms, exactly half decay in one half-life. A bigger sample decays MORE atoms per second (higher activity), but the TIME to lose half stays the same. This is because decay is a fraction-based (exponential) process, not a fixed number per second.

After 2 half-lives, is the whole sample gone?

No, this is the most common mistake. After 1 half-life, half is left (1/2). After 2 half-lives, half of that half is left, so 1/4 remains, not zero. Half-lives do not add up to finish the sample. Fraction left after n half-lives = (1/2) to the power n. The sample never fully reaches zero in theory.

Is half-life the same as mean life?

No. Mean life (tau) is the AVERAGE life of all nuclei and is longer than half-life. The link is tau = 1/lambda and T = 0.693/lambda, so T = 0.693 x tau, meaning half-life is about 69.3 percent of mean life. Half-life = time for half to go; mean life = average survival time.

Where does the number 0.693 come from?

It is ln 2 (natural log of 2). Start from the decay law N = N0 e^(-lambda t). At one half-life, N = N0/2. So 1/2 = e^(-lambda T). Taking natural log: ln(1/2) = -lambda T, which gives lambda T = ln 2 = 0.693. So T = 0.693/lambda. The 2 appears because we are halving.

Do temperature, pressure or chemical state change half-life?

No. Radioactive decay comes from inside the nucleus, so it is not affected by outside conditions like heat, pressure, or whether the atom is in a compound. Half-life stays constant for a given isotope. This makes it useful for carbon dating and finding the age of rocks.

⚠️ The NEET trap
✗ Thinking the sample is fully gone after 2 half-lives, or that half-life doubles if you double the number of atoms.
✓ After n half-lives the fraction LEFT is (1/2)^n: 1/2, 1/4, 1/8, 1/16... it never hits zero. Half-life is fixed and independent of initial amount.
🧠 NTA loves to slip in 'after how many half-lives is 1/8 (or 6.25 percent) left?' - count the halvings: 1/8 = 3 half-lives, 6.25% = 1/16 = 4 half-lives.
Next concept: Difference Between Mean Life and Half-Life →Keep learning — 2 minFeeling ready? Solve the nuclei NEET PYQs ›Or practice on your phone — get the free MedicNEET app ›

Frequently asked

What is the formula for half-life?

T = 0.693 / lambda, where lambda is the decay constant. Also T = 0.693 x tau, where tau is the mean life.

What is the SI unit of half-life?

The second (s). In practice half-lives are quoted in seconds, minutes, days, or years depending on the isotope.

How much of a sample is left after 3 half-lives?

Fraction left = (1/2)^3 = 1/8, which is 12.5 percent. The rest (7/8) has decayed.

Does half-life depend on the decay constant?

Yes, inversely. A larger decay constant lambda means a shorter half-life (faster decay), since T = 0.693/lambda.

Is half-life the same for all radioactive elements?

No. Each isotope has its own fixed half-life, ranging from tiny fractions of a second to billions of years (for example, uranium-238 has a half-life of about 4.5 billion years).

Related concepts:
Difference Between Mean Life and Half-LifeWhat Is Decay Constant (lambda) and Its MeaningHalf-Life Numericals: Fraction Remaining After n Half-LivesRadioactive Decay Law: N = N0 e^(-lambda t) DerivationWhat Is Activity of a Radioactive Sample?
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